
theorem Th15:
  for n being Nat holds for x1,x2 being Element of
dyadic(n) st x1 < x2 holds x1 <= (axis(x2)-1)/(2|^n) & (axis(x1)+1)/(2|^n)
  <= x2
proof
  let n be Nat;
  let x1,x2 be Element of dyadic(n);
  assume
A1: x1 < x2;
  then axis(x1) < axis(x2) by Th14;
  then
A2: axis(x1) + 1 <= axis(x2) by NAT_1:13;
  axis(x1) < axis(x2) by A1,Th14;
  then axis(x1) + 1 <= axis(x2) by NAT_1:13;
  then
A3: axis(x1) <= axis(x2) - 1 by XREAL_1:19;
A4: (axis(x1)+ 1)/(2|^n) <= axis(x2)/(2|^n) by A2,XREAL_1:72;
  axis(x1)/(2|^n) <= (axis(x2) - 1)/(2|^n) by A3,XREAL_1:72;
  hence thesis by A4,Th10;
end;
